by Okamoto [1]} (which can be applied since \(\frac{d}{N-1}-\frac{1}{2}>0\) ). It follows that
\( P_{d,N} ={\mathbb {P}}(\xi _{N-1}\le d-1)\ge 1-{\rm exp}\,\left(-\left(\delta -\frac{1}{2}\right)^2(N-1)\right)\rightarrow 1\) 
